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6
votes
1answer
232 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
2
votes
1answer
143 views

Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
5
votes
1answer
228 views

rationality question while dealing with an isogeny

I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment ! So here is the situation. Let ...
7
votes
2answers
324 views

Forms of algebraic varieties

Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
5
votes
1answer
232 views

Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. Is it true that ...
10
votes
0answers
263 views

Unramified Galois cohomology of number fields

I'm trying to understand unramified Galois cohomology of number fields a bit better. Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...
4
votes
1answer
160 views

What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$ The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...
5
votes
0answers
91 views

References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response. What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...
1
vote
1answer
240 views

Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
1
vote
1answer
129 views

$L/k$ forms for affine schemes of finite type

Notations and terminology: Let $k$ be a field and $X$ be a $k$-scheme. Denote by $X_L$ the scheme $X\times_k\rm Spec(L)$. For a field extension $L/k$, a $L/k$ form is a $k$-scheme $Y$ such that there ...
0
votes
0answers
145 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
3
votes
1answer
67 views

Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$; ...
0
votes
0answers
132 views

Descent for group actions

Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$. Finally, suppose I have an action $\sigma$ of $G$ on a ...
3
votes
0answers
82 views

non-Abelian inflation-restriction sequence?

Let $1 \to H \to G \to G/H \to 1$ be a group extension. The long exact sequence that arises from Hochschild-Serre spectral sequence for this extension relates objects that are clasified by cohomology ...
6
votes
2answers
561 views

Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...
3
votes
2answers
203 views

Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success. Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...
2
votes
1answer
127 views

the sixth morphism in the long exact sequence associated to the Hochschild-Serre spectral sequence

The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is $$ \begin{array}[t]{lll} 1 \to & H^1(G/H, A^{G/H}) ...
3
votes
2answers
244 views

Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?
2
votes
0answers
74 views

do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...
1
vote
1answer
123 views

gluing gerbes over a spectrum of a field

A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar statements hold in other ...
4
votes
2answers
231 views

Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). Since the cited ...
4
votes
0answers
158 views

geometric interpretation of the transgression map

Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G$, so ...
2
votes
0answers
239 views

Where does the name Euler System come from?

I've recently been reading about Euler systems, and was curious where the name comes from. In particular, while the notion of an Euler system is still not rigorously defined, does the idea resemble ...
2
votes
0answers
201 views

galois cohomology over finite field

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$. Do we have a formula to ...
3
votes
1answer
269 views

Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...
1
vote
1answer
110 views

action of automorphisms on the Galois cohomology of the function field of a variety

Let $C$ be a (quasi-projective) variety over an algebraically closed field $k$ and let $k(C)$ be its field of rational functions. Then for any birational map $\sigma: C \dashrightarrow C$ there is an ...
1
vote
1answer
152 views

Is the number of twists of a curve with a section in a given field finite

Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$. Is the number of twists of $X$ always infinite? (The answer is no, because there aren't any ...
0
votes
1answer
195 views

surjectivity of rational points induced by surjective map from affine space

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) ...
1
vote
1answer
376 views

Cohomology of Groups at Gregory Berhuy's Book

Regarding Gregory Berhuy's book "An Introduction to Galois Cohomology and its Applications": The book defined a cohomology sets for non-abelian $G$-groups. Let $A$ be a $G$-group, we define a ...
2
votes
1answer
171 views

What's the minimum number of generators for the wild inertia?

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...
11
votes
1answer
525 views

Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...
1
vote
1answer
220 views

Biprincipal spaces and their composition, following Serre's “Galois Cohomology”

When reviewing my notes and Serre's book "Galois Cohomology" Chapter 5 dealing with non-abelian group cohomology, I realized that I don't fully understand the concept of biprincipal spaces such as ...
1
vote
1answer
274 views

Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...
16
votes
1answer
705 views

Leopoldt's conjecture and cup-products

Among the many equivalent formulations of Leopoldt's conjecture, this one is probably the shortest: For any number field $K$, prime number $p$, finite set $S$ of primes of $K$ containing the primes ...
4
votes
2answers
588 views

Galois cohomology H^1(Q_p, Z_p(2)) = 0?

For Tate twists Z_p(2), which is defined by the projective limit of \mu_{p^m}(2) over all m>0, I would like to calculate H^1(Q_p, Z_p(2)). I guess this is zero, but cannot prove it. Is it possible ...
1
vote
0answers
135 views

Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?

Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...
0
votes
3answers
387 views

Brauer group of complete DVR

Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$. Let $K_{nr}$ be the maximal unramified extension of $K$ and let $A_{nr}$ be its ring of ...
1
vote
1answer
228 views

a question on continuity of $G$-module for a profinite group $G$

I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a ...
0
votes
0answers
213 views

Ordered Cech(-like) complexes that compute etale cohomology (of fields!)

It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...
5
votes
1answer
304 views

Equality of Galois modules

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module. One can define two subgroups of the Galois cohomology group $H^i(k,M)$: the group of elements of $H^i(k,M)$ ...
6
votes
1answer
910 views

p-adic Hodge theory for varieties defined over \C _p ?

I have a question on p-adic Hodge theory: When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...
2
votes
1answer
294 views

Additive form of Hilbert 90 for schemes?

First, I am by no means well-versed on cohomology so I apologize if this is too elementary. I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of ...
0
votes
1answer
376 views

Poitou-Tate dualities for Galois representations into power series rings?

Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are ...
2
votes
2answers
327 views

Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?

For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...
6
votes
1answer
544 views

Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$. When is $$ ...
19
votes
2answers
1k views

How is etale cohomology of integer rings related to Galois cohomology?

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups $H^i_{\mathrm{et}}(\operatorname{Spec} ...
3
votes
1answer
708 views

Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology? Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$. What if $G = ...
10
votes
2answers
723 views

Compatibility of Bloch-Kato and Beilinson-Bloch

Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$ be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture ...
11
votes
1answer
545 views

Is there a canonical height on the Weil-Chatelet group?

Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the ...
7
votes
1answer
349 views

Is H^2(W_p,C^times) well-known?

Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated? (This group ...